Let $\phi_{n}(x)$ be the $n$-th cyclotomic polynomial. What are the restrictions to $n$ (if any) to have $\phi_{n}(x)$ divides $\phi_{2n}(x)$ (where division is in $\mathbb{Z}[x]$)?Or is it true that $\frac{\phi_{2n}(x)}{\phi_{n}(x)}\in\mathbb{Z}[x]$ for all integers $n$?

well, $\phi_{2011}$ and $\phi_{4022}$ have the same degree.
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Xandi TuniMar 24 '11 at 13:10

Maybe I should think a bit before writing... Of course those polynomials will very often have the same degree (as soon as n is odd)... Kikiriku's answer is much better and does not use the irreducibility of those polynomials...
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AurelienMar 25 '11 at 13:35